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An OpenAI model has disproved a central conjecture in discrete geometry

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Re: An OpenAI model has disproved a central conjecture in discrete geometry

#301

Post by cpard »

doubledamio wrote: post #238
cpard wrote: post #224

The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…

I’ve always been skeptical about the role of LLMs in mathematics, but this is the first time I’ve seen this argument, and I actually find it very compelling. Maybe LLMs will help us develop more horizontal understanding of the field.

It's up to us I think. We can use LLMs to generate web pages in candy crash style and end up dumper by outsourcing thinking to the machines or we can use it to expand our cognitive capabilities.

What makes me more of an optimist in this case is that people who today decide to go into these sciences are mostly people who are driven by intellectual activity so I feel they are the right ones to figure this out, probably more so than us the engineers.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#302

Post by Nevermark »

symfrog wrote: post #260
bbor wrote: post #246

Earlier quoted context omitted.

To keep my usual rant short: I think you’re assuming a categorical distinction between those two types of innovations that just doesn’t exist. Calculus certainly required some fundamental paradigm shifts, but there’s a reason that they didn’t have to make up many words wholesale to explain it! Also we shouldn’t be thinking about what LLMs are good at, but rather what any computer ever might be good at. LLMs are alrea…

We have had LLMs for much longer than 3 years.

I took humans thousands of years, then hundreds of years, to come to terms with very basic concepts about numbers.

Its amazing to me when people talk about recombining things, or following up on things as somehow lesser work.

People can't separate the perspective they were given when they learned the concepts, that those who developed the concepts didn't have because they didn't exist.

Simple things are hard, or everything simple would have been done hundreds of years ago, and that is certainly not the case. Seeing something others have not noticed is very hard, when we don't have the concepts that the "invisible" things right in front of us will teach us.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#303

Post by nextaccountic »

interroboink wrote: post #240
daishi55 wrote: post #202

Earlier quoted context omitted.

> the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept. Isn't this just anthropocentrism? Why is understanding only valid if a human does it? Why is knowledge only for humans? If another species resolved the contradictions be…

It's a bit of an "if a tree falls in the forest but nobody hears it, does it make a sound?" quandary. Sure, maybe some aliens in a distant galaxy understand quantum mechanics better than we do. That's great, but it has no bearing on our little bubble of existence. Though perhaps more to your point, if some superhuman AI is developed, and understands things better than us without telling us about it (or being unable t…

It's less likely that aliens of distant galaxies will appreciate this rather than, you know, AI themselves

We are in the birth of the AI age and we don't know how it will look like in 100 or 1000 or 10000 or 100000 years (all those time frames likely closer than possible encounters with aliens from distant galaxies). It's possible that AI will outlast humans even

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#304

Post by bwfan123 »

midtake wrote: post #291
m-hodges wrote: post #40

To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…

You have a good point about the human rate of mathematical discovery, but Ayer was an idiot and later Witt contradicted early Witt. For the "already implicit" claim to be true, mathematics would have to be a closed system. But it has already been proven that it is not. You can use math to escape math, hence the need for Zermelo-Frankel and a bunch of other axiomatic pins. The truth is that we don't really understand…

> Whether that greater math has the same seemingly closed system properties is not something that can be known

negative numbers were invented to solve equations which only used naturals. irrationals were invented to solve equations which could be expressed with rationals. complex numbers were invented to represent solutions to polynomials. so on and so forth. At each point new ideas are invented to complete some un-answerable questions. There is a long history of this. Any closed system has unanswerable questions within itself is a paraphrasing of goedel's incompleteness theorem.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#305

Post by kelseyfrog »

sumeno wrote: post #282
kelseyfrog wrote: post #278

Earlier quoted context omitted.

> I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus. (I'm open to being proven wrong.) The experiment is feasible. If it were performed and produced a positive result, what would it imply/change about how you see LLMs?

How are you going to train a frontier level llm with no references to post 1700 mathematics?

Time cutoff LLMs are regularly posted to HN. It takes just one success to prove feasibility.

Besides, we can forecast our thoughts and actions to imagined scenarios unconditioned on their possibility. Something doesn't have to be possible for us to imagine our reactions.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#306

Post by dilap »

dawnerd wrote: post #299
raincole wrote: post #280

I like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)

Yet it still codes like a junior developer that memorized all of stack overflow.

Personally I don't find this to be true anymore! It's not always great and does still will often tend towards unneeded complexity (especially if not pushed a bit), but I often find GPT 5.5 writing code I would have written myself. This was very much not true with earlier models (who make something that worked, but I'd always have to rewrite to make it "good code").

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#307

Post by KalMann »

atleastoptimal wrote: post #206

To all AI skeptics: What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? If we compare AI now vs 2022 the difference is outstandingly stark. Do you believe this improvement will just stop before it eclipses all humans in everything we care about?

I think there's been natural but steady progress with since 2024 with the release of the o1 model, which showed impressive reasoning capabilities. But I think it's wrong to look at the magnitude of the accomplishments and assume that will be field independent. We don't know the range of problems reasoning techniques are useful for. What we see here is refinement of capabilities that have been noticeable for years.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#308

Post by raincole »

dawnerd wrote: post #299
raincole wrote: post #280

I like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)

Yet it still codes like a junior developer that memorized all of stack overflow.

PhDs code like that too. Especially if they're statisticians :)

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#309

Post by alsetmusic »

> AI is about to start taking a very serious role in the creative parts of research, and most importantly AI research itself. While this progress is not unexpected, it reinforces the urgency we feel about understanding this next phase of AI development, the challenges of aligning very intelligent systems, and the future of human-AI collaboration.

I find this hyperbolic, but ya gotta juice up the upcoming IPO. I hate that they took an interesting announcement and reminded me why I hate tech and our society at the end.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#310

Post by Nevermark »

voooduuuuu wrote: post #125

Ask an LLM to invent a new word and post it here. You will see that it simply combines words already in the training data.

You must be joking? Unless by combining words you mean digging deep into Latin and Greek etymology, finding something pithy and linguistically associative.

I can assure you, the percentage of people who can do what they do when it comes to crafting terms, and related sets of terms, for nuanced and novel ideas is very very small.

It happens this is something I do nearly every day.

Models respond to the level of dialogue you have with them. Engage with an informed perspective on terminological issues and they respond with deep perspectives.

I am routinely baffled at the things people say models can't do, that they do effortlessly. Interaction and having some skill to contribute helps here.

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